Modular elliptic directions with complex multiplication (with an application to Gross’s elliptic curves)

Abstract

Let $A_f$ be the abelian variety attached by Shimura to a normalized newform $f\in S_2({\Gamma }_1(N))$ and assume that $A_f$ has elliptic quotients. The paper deals with the determination of the one dimensional subspaces (elliptic directions) in $S_2({\Gamma }_1(N))$ corresponding to the pullbacks of the regular differentials of all elliptic quotients of $A_f$. For modular elliptic curves over number fields without complex multiplication (CM), the directions were studied by the authors in [8]. The main goal of the present paper is to characterize the directions corresponding to elliptic curves with CM. Then we apply the results obtained to the case $N=p^2$, for primes $p>3$ and $p\equiv 3$ mod $4$. For this case we prove that if $f$ has CM, then all optimal elliptic quotients of $A_f$ are also optimal in the sense that its endomorphism ring is the maximal order of $\mathbb{Q}(\sqrt{-p})$. Moreover, if $f$ has trivial Nebentypus then all optimal quotients are Gross’s elliptic curve $A(p)$ and its Galois conjugates. Among all modular parametrizations $J_0(p^2)\to A(p)$, we describe a canonical one and discuss some of its properties.